Paris Law

The Paris Law is an empirical equation used to describe the rate of fatigue crack growth in materials.  It relates the rate of crack growth to the stress intensity factor range per cycle.  The Paris Law is widely used in fatigue analysis and is particularly valuable for predicting the fatigue life of materials subjected to cyclic loading conditions.

 

Paris Law Formula
\( da\;/\;dN \;=\; C \; (\Delta K)^m \)
Symbol	English	Metric
\( da\;/\;dN \) = Crack Growth Rate (How Much the Crack in a Material Will Grow per Cycle)	\(in\)	\(mm\)
\( \Delta K \) = Stress Intensity Factor Range (Which is the Difference Between the Maximum and Minimum Values of the Stress Intensity Factor During Each Loading Cycle)	\(Kpsi \; \sqrt{in}\)	\(MPa \; \sqrt{m}\)
\( C and m \) = Material Constants (Depend on the Specific Material and Loading Conditions)	\(dimensionless\)	\(dimensionless\)

  

stress intensity factor Formula
\( K \;=\; Y \; \sigma \; \sqrt{ \pi \; a }  \)
Symbol	English	Metric
\( K \) = Stress Intensity Factor	\(Kpsi \; \sqrt{in}\)	\(MPa \; \sqrt{m}\)
\( Y \) = Geometric Parameter that Depends on the Specimen’s Geometry and the Crack's Location	\(in\)	\(mm\)
\( \sigma \) = Applied Stress	\(lbf \;/\; in^2\)	\(Pa\)
\( \pi \) = Pi	\(3.141 592 653 ...\)	\(3.141 592 653 ...\)
\( a \) = Crack Length	\(in\)	\(mm\)

 

The Paris Law essentially states that the crack growth rate is proportional to a power of the stress intensity factor range. The exponent $$m\$$ and the constant $$C\$$ are determined experimentally for a given material under specific loading conditions.  These constants may vary depending on factors such as material type, microstructure, temperature, and loading frequency.

Engineers use the Paris Law to predict the rate of fatigue crack growth and estimate the remaining fatigue life of structures subjected to cyclic loading.  By monitoring crack growth rates and applying the Paris Law, engineers can assess the integrity of components and schedule maintenance or replacement before catastrophic failure occurs.  The Paris Law has been validated through extensive experimental testing across a wide range of materials and loading conditions, making it a valuable tool in fatigue analysis and design.